Reply to the Comment on "Shell-Shaped Quantum Droplet in a Three-Component Ultracold Bose Gas"

Abstract

In our Letter (Phys. Rev. Lett. 134, 043402 (2025)), we proposed a self-bound shell-shaped BEC in a three-component ($1,2,3$) Bose gas, where $(2,3)$ and $(1,2)$ droplets are linked as core-shell structure. A recent Comment (Ancilotto, 2505.16554) argued that a ``dimer" configuration should be instead the ground state, where $(2,3)$ and $(1,2)$ stay side-by-side. Moreover, Ancilotto also explored the state formation, finding that a naive trap-release protocol was unable to produce the core-shell structure. In this reply we show that our core-shell structure is an excited state for finite-size systems, while it becomes energetically degenerate with dimer configuration in thermodynamic limit. Furthermore, we find the core-shell structure is locally stable under external perturbations, and if one pays careful attention to mode-matching, a trap-release protocol can well produce this structure.

Figures (3)

• Figure 1: Density profiles of core-shell (a1,a2,a3) and dimer (b1,b2,b3) states. The atoms numbers are $(N_1,N_2,N_3)/10^5=(0.3,0.56,0.03)$ in (a1,b1), $(1,1.76,0.03)$ in (a2,b2) and $(2,3.47,0.03)$ in (a3,b3). (a1,a2,a3) are for radial densities, and (b1,b2,b3) are densities at $x=y=0$. The relative energy difference, defined as $\delta E\equiv (E_{\rm c-s}-E_{\rm dimer})/|E_{\rm c-s}|$, decreases as the shell atom number grows: $\delta E=13.36\%$(a1,b1), $6.0\%$(a2,b2) and $3.12\%$(a3,b3). Here we consider a realistic $^{23}$Na-$^{39}$K-$^{41}$K ('1'-'2'-'3') mixture near $B\sim 150$G with $a_{23}=-200a_0$.
• Figure 2: Deformation of core-shell structure under a magnetic field gradient $B'=0.08 E_0/l_0$ ($l_0=1\mu m$ and $E_0=\hbar^2/(m_Kl_0^2)$). Here we take the initial core-shell state as in Fig.2(a3) of Ma. To clearly see the deformation we just plot out $n_1$ (shell component) and $n_3$ (core component) at $y=0$; $n_2$ (not shown here) exhibits similar core-shell structure.
• Figure 3: Time evolution of radial densities after the system is released from an isotropic harmonic trap. Here we take the trap frequency as $\omega_{\rm K}=5KHz$, and $\omega_{\rm Na}=\omega_{\rm K}\sqrt{m_{\rm K}/m_{\rm Na}}$; the atom numbers are $(N_1,N_2,N_3)/10^5=(1,1.86,0.1)$, and the time is $t\ (ms)=0$(a), $0.15$(b) and $0.3$(c). Dashed lines with according color show the densities of equilibrium core-shell state in free space. The interaction parameters are the same as in Fig.\ref{['fig_compare']}.